Skip to main content Accessibility help
×
Home

On Two Damage Accumulation Models and Their Size Effects

  • F. Ballani (a1), D. Stoyan (a1) and S. Wolf (a1)

Abstract

Two cumulative damage models are considered, the inverse gamma process and a composed gamma process. They can be seen as ‘continuous’ analogues of Poisson and compound Poisson processes, respectively. For these models the first passage time distribution functions are derived. Inhomogeneous versions of these processes lead to models closely related to the Weibull failure model. All models show interesting size effects.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On Two Damage Accumulation Models and Their Size Effects
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On Two Damage Accumulation Models and Their Size Effects
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On Two Damage Accumulation Models and Their Size Effects
      Available formats
      ×

Copyright

Corresponding author

Postal address: Institute of Stochastics, TU Bergakademie Freiberg, Prüferstrasse 9, D-09596 Freiberg, Germany.
∗∗ Email address: ballani@mi.fu-berlin.de

References

Hide All
Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press.
Bažant, Z. P. and Planas, J. (1998). Fracture and Size Effect in Concrete and other Quasibrittle Materials. CRC Press, Washington, DC.
Çinlar, E. (1980). On a generalization of gamma processes. J. Appl. Prob. 17, 467480, 893.
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. 1, 2nd edn. Springer, New York.
Dufresne, F., Gerber, H. U. and Shiu, E. S. W. (1991). Risk theory with the gamma process. ASTIN Bull. 22, 177192.
Duxbury, P. M. and Leath, P. L. (1994). Exactly solvable models of material breakdown. Phys. Rev. B 49, 1267612686.
Evans, M., Hastings, N. and Peacock, B. (1993). Statistical Distributions, 2nd edn. John Wiley, New York.
Ferguson, T. S. (1974). Prior distributions of spaces of probability measures. Ann. Statist. 2, 615629.
Ferguson, T. S. and Klass, M. J. (1972). A representation of independent increment processes without Gaussian components. Ann. Math. Statist. 43, 16341643.
Gautschi, W. (1998). The incomplete gamma functions since Tricomi. In Tricomi's Ideas and Contemporary Applied Mathematics (Atti dei Convegni Lincei 147), Accademia Nazionale dei Lincei, Roma, pp. 203237.
Jeulin, D. (1994). Random structure models for composite media and fracture statistics. In Advances in Mathematical Modelling of Composite Materials, ed. Markov, K., World Scientific, Singapore, pp. 239289.
Kingman, J. F. C. (1993). Poisson Processes (Oxford Studies Prob. 3). Oxford University Press.
Krajcinovic, D. (1996). Damage Mechanics, Applied Mathematics and Mechanics. Elsevier, Amsterdam.
Mann, N. R., Schafer, R. E. and Singpurwalla, N. D. (1974). Methods for Statistical Analysis of Reliability and Life Data. John Wiley, New York.
Onar, A. and Padgett, W. J. (2000). Inverse Gaussian accelerated test models based on cumulative damage. J. Statist. Comput. Simul. 64, 233247.
Park, C. and Padgett, W. J. (2005). New cumulative damage models for failure using stochastic processes as initial damage. IEEE Trans. Reliab. 54, 530540.
Pompe, W. et al. (1985). Mechanical Properties of Brittle Materials, Modern Theories and Experimental Evidence. Elsevier, Amsterdam.
Rolski, T., Schmidli, H., Schmidt, H. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, New York.
Sobczyk, K. (1987). Stochastic models for fatigue damage of materials. Adv. Appl. Prob. 19, 652673.
Sobczyk, K. (1997). On cumulative Jump models for random deterioration processes. Ann. Univ. Maria Curie-Sklodowska, Lublin 15, 145157.
Sobczyk, K. and Spencer, B. F. (1992). Random Fatigue: From Data to Theory. Academic Press, Boston, MA.
Todinov, M. T. (2002). Statistics of defects in one-dimensional components. Comput. Mat. Sci. 24, 430442.
Wolf, S., Wiegand, S., Stoyan, D. and Walther, H. B. (2005). The compressive strength of AAC – a statistical investigation. In Autoclaved Aerated Concrete. Innovation and Design, Taylor and Francis, London, pp. 287295.

Keywords

MSC classification

On Two Damage Accumulation Models and Their Size Effects

  • F. Ballani (a1), D. Stoyan (a1) and S. Wolf (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed